On a multiphase multicomponent model of biofilm growth

Analysis of a mathematical model of rheumatoid arthritis

Rheumatoid arthritis is an autoimmune disease characterized by inflammation in the synovial fluid within the synovial joint connecting two contiguous bony surfaces. The inflammation diffuses into the cartilage adjacent to each of the bony surfaces, resulting in their gradual destruction. The interface between the cartilage and the synovial fluid is an evolving free boundary. In this paper we consider a two-phase free boundary problem based on a simplified model of rheumatoid arthritis. We prove global existence and uniqueness of a solution, and derive properties of the free boundary. In particular it is proved that the free boundary increases in time, and the cartilage shrinks to zero as [Formula: see text], even under treatment by a drug. It is also shown in the reduced one-phased problem, with cartilage alone, that a larger prescribed inflammation function leads to a faster destruction of the cartilage.

One-domain approach for studying multiphase transport phenomena in biofilm growing systems

The one-domain approach (ODA) was used as an alternative to solve fluid-biofilm interfacial behavior in a 2-D model for diffusion-reaction-convection coupled with prediction of irregular growth of biofilms via a cellular automaton strategy. The simulations exhibited errors of <7% compared with the porosity of a previously reported capillary experimental system. Additionally, biofilm surface geometrical aspects were satisfactorily compared with reports of experimental and similar rigorously simulated benchmark systems. The method developed was applied to simulate typical biofilm systems predicting recirculation flow patterns, interface concentration profiles, and clogging of the inlet section of the capillary tube, which are phenomena that affect the efficiency of diverse biotechnological applications, including membrane bioreactors and biofilters. The ODA method applied to the governing equations of momentum and mass transfer combined with a cellular automaton algorithm is a suitable and straightforward approach for modeling solid-state fermentation at different sophistication levels.

A Mixed-Culture Biofilm Model with Cross-Diffusion

We propose a deterministic continuum model for mixed-culture biofilms. A crucial aspect is that movement of one species is affected by the presence of the other. This leads to a degenerate cross-diffusion system that generalizes an earlier single-species biofilm model. Two derivations of this new model are given. One, like cellular automata biofilm models, starts from a discrete in space lattice differential equation where the spatial interaction is described by microscopic rules. The other one starts from the same continuous mass balances that are the basis of other deterministic biofilm models, but it gives up a simplifying assumption of these models that has recently been criticized as being too restrictive in terms of ecological structure. We show that both model derivations lead to the same PDE model, if corresponding closure assumptions are introduced. To investigate the role of cross-diffusion, we conduct numerical simulations of three biofilm systems: competition, allelopathy and a mixed system formed by an aerobic and an anaerobic species. In all cases, we find that accounting for cross-diffusion affects local distribution of biomass, but it does not affect overall lumped quantities such as the total amount of biomass in the system.

Free boundary problems in biology

In this paper, I review several free boundary problems that arise in the mathematical modelling of biological processes. The biological topics are quite diverse: cancer, wound healing, biofilms, granulomas and atherosclerosis. For each of these topics, I describe the biological background and the mathematical model, and then proceed to state mathematical results, including existence and uniqueness theorems, stability and asymptotic limits, and the behaviour of the free boundary. I also suggest, for each of the topics, open mathematical problems.